Large A.C. Machines (eBook)

Boguslawsky I. Z. is a professor in the Theoretical Electrical Engineering and Electromechanic Department at Peter the Great St. Petersburg Polytechnic University and is a senior member of IEEE. He has spent 50 years in the electric machine-building industry (calculation and design of large alternating current machines). His work at the “Elektrosila” Works Stock Company was in power machines and the Leningrad Electrical Machine Building Plant.Korovkin N. V. is a professor in and head of the Theoretical Electrical Engineering and Electromechanic Department at Peter the Great St. Petersburg Polytechnic University. He has more than 35 years’ experience in power engineering.Hayakawa M. is an emeritus professor of the University of Electro-Communications (UEC), a fellow of The Institute of Electronics, Information and Communication Engineers (IEICE), Japan, and a senior member of The Institute of Electrical Engineers (IEE), Japan. He is the CEO of the Hayakawa Institute of the Seismo Electromagnetics Co. Ltd. (UEC Incubation Center) and a visiting professor at the Advanced Wireless & Communications Research Center (AWCC), UEC, Tokyo. He has more than 45 years' experience in electromagnetics. The authors gratefully welcome any comments and suggestions (in English, German, or Russian languages) on the contents of this monograph. They should be addressed to Polytechnicheskaya str., 29, Peter the Great St. Petersburg Polytechnic University, Theoretical Engineering and Electromechanic Department, St. Petersburg, 195251, Russia; email: Nikolay.korovkin@gmail.com.

Preface 5
Acknowledgments 7
Contents 9
About the Authors 27
1 Problems Formulation 28
1.1 Requirements to High-Power A.C. Motors and Generators 28
1.2 Actual Problems of Modern Electric Machine-Building Industry Their Solution in the Monograph
1.3 Presentation Order of Monograph Contents and Structure 34
1.3.1 Choice of Investigation Methods: Position of Monograph’s Authors 34
1.3.2 Presentation Order of Monograph Contents 35
1.3.3 Checking Methods Developed in the Monograph 36
1.3.4 Symbols Accepted in the Monograph 36
1.4 Order of References 37
2 Investigation Methods of Performance Characteristics for Double-Fed Machines with Converter in Rotor Circuit. Summary of Main Investigation Stages of A.C. Machines 39
2.1 Peculiarities of Investigation Methods 39
2.2 Problem Statement 41
2.3 Frequencies and Amplitudes of Voltage and Current First Harmonics in Machine Rotor and Stator Windings 42
2.3.1 Ratio of Frequencies {{/varvec f}}_{{{{/varvec ROT}},1}} /,{{/varvec and}}/,{{/varvec f}}_{{{{/varvec ST}},1}} /,{{/varvec at}}/,{{/varvec n}}_{{{{/varvec REV}}}} = {{/varvec var}} 42
2.3.2 Ratio of Voltages UROT,1 and Ust,1 at nREV = var 43
2.4 Frequencies and Amplitudes of Voltage and Current Higher Time Harmonics in Machine Rotor and Stator Windings 44
2.4.1 Ratio of Frequencies {{/varvec f}}_{{{{/varvec ROT}}{{/varvec ,}}{{/varvec Q}}}} /,{{/varvec and}}/,{{/varvec f}}_{{{{/varvec ST}}{{/varvec ,}}{{/varvec Q}}}} /,{{/varvec at}}/,{{/varvec n}}_{{{{/varvec REV}}}} = {{/varvec var}} 45
2.4.2 Ratio of Voltages {{/varvec U}}_{{{{/varvec ROT}}{{/varvec ,}}{{/varvec Q}}}} {{/varvec U}}_{{{{/varvec ST}}{{/varvec ,}}{{/varvec Q}}}} /,{{/varvec at}}/,{{/varvec n}}_{{{{/varvec REV}}}} = {{/varvec var}}
2.5 Method for Solving Both Problems Two Systems of Equations
2.5.1 Magnetization Characteristics Presentations {{/varvec /uptheta}}_{0.1} = {{/varvec /uptheta}}/left( {{/hbox{F}}_{{{{/rm M}}.{{/rm C}}.,1}} } /right) in Piecewise Linear Function Form 47
2.5.2 Peculiarities of Solving Both Systems 48
2.6 Check of Methods 49
2.7 Excitation System Peculiarities 50
2.8 Summarizing the Results: Main Stages of A.C. Machine Investigations with Rotor Short Circuited Windings 50
2.8.1 Stator and Rotor Circuits Frequency Voltage. Rotational Speed of Rotor and Stator Fields in Air Gap 50
2.8.2 Ampere’s Law Equations 51
2.8.3 Kirchhoff’s Second Law Equations for Stator Winding 51
2.8.4 Kirchhoff’s Second Law Equations for Rotor Loops. System of Equations 52
Appendix 2.1 52
A.2.1.1ƒDFM Rotor Current and MMF Under Load 53
A.2.1.2 Phase Angle Defining Complex Amplitudes Position of MMF {{/bf F}}_{{{{/bf ROT}},1}} and Current {{/bf I}}_{{{{/bf ROT}},1}} (in Stator Coordinates) 55
A.2.1.3 Rotor Winding Voltage, Its Components 55
A.2.1.4 Phase Angle Defining Complex Amplitude Position UROT,1 56
A.2.1.5 Rotor Winding Power Factor Active and Reactive Winding Power
A.2.1.6 DFM Rotor Winding Design Peculiarities 56
Appendix 2.2 57
Brief Conclusions 58
References 59
I. Monographs, General Courses, Textbooks 59
II. Induction Machines. Papers, Inventor’s Certificates 60
III. State Standards (IE?, GOST and so on) 61
3 Stator MMF Harmonics at Non-sinusoidal Machine Power Supply (for M ? 1, Q ? 1) 62
3.1 Initial Data and Its Representation 62
3.2 Stator Winding Design Peculiarities Its Number of Phases {{/bf m}}_{{{{/bf PH}}}}
3.3 MMF Harmonics at mPH = 3 65
3.4 MMF Harmonics at {{/bf m}}_{{{{/bf PH}}}} = {{/bf 6}} 68
3.5 MMF Harmonic Comparison at mPH = 3 and mPH = 6 70
3.6 EMF Frequency in Magnetically Coupled Loops (mEL ? 1, Q ? 1) 72
3.6.1 Salient Pole Machines Operating in Synchronous Speed Modes 72
3.6.2 A.C. Machines in Asynchronous Modes /left( {{{/varvec /omega}}_{{{{/varvec REV}}}} /ne /frac{{{{/varvec /omega}}_{ 1} }}{{{/varvec p}}}} /right) 76
3.7 MMF Harmonics of Magnetically Coupled Loops in Multiphase Stator Winding at Non-sinusoidal Power Supply and their Representation 78
3.8 Magnetically Coupled Loops in Machine Operating with Nonlinear Network Elements 79
Appendix 3.1 81
Brief Conclusions 82
References 83
I. Monographs, general courses, textbooks 83
II, III. Asynchronous and synchronous machines. Papers, inventor’s certificates, patents 84
IV. State Standards (IEC, GOST and so on) 84
4 Peculiarities of Currents Investigation in Magnetically Coupled Circuits for A.C. Machines with Short-Circuited Rotor Windings 85
4.1 General 86
4.2 The Problem Determination and Algorithm of Solution 87
4.3 Method Investigating of Operation Characteristics of Powerful Squirrel Cage Motors with Nonlinear Parameters 92
4.4 Experimental Data and Calculating Results Comparison for the Torque M on the Shaft Depended on Slip SSL for DAZ-Type Motor 96
4.5 Generalized Characteristic of Rotor Current in Cage Elements 97
Appendix 4.1: Screen Calculation Method of Large Low-Frequency Motor Pole Shoe 99
A.4.1.1 Simply Connected Domain 100
A.4.1.2 Multiply Connected Domain 108
A.4.1.3 Irregular Grid with Arbitrary Configuration Elements 109
A.4.1.4 The Screen Element D.C. Resistance Depends on the Temperature Distributed in It (Nonlinear Problem) the Domain Is Simply—Or Multiply Connected
A.4.1.5 Screen Element Reactance Accounting 110
Appendix 4.2: Determination of Average Flux Density Point Location by “Mean Value Theorem (Lagrange)” [2, 3] 111
Appendix 4.3: Equivalent Circuit of Powerful Induction Motors Operating in Nonlinear Networks 111
A.4.3.1 Basic Equations of Powerful Asynchronous Machine Substitution Patterns /left( {{{/bf Q}}_{{{{/bf TIM}}}} /ge {{/bf 1}}} /right) 112
A.4.3.2 Machine Circuit Losses Components and Shaft Power 114
A.4.3.3 Additional Losses and Equivalent Circuit Impedances 116
Brief Conclusions 117
References 120
I. Monographs, General Courses, Textbooks 120
II. Induction Machines. Papers, Inventor’s Certificates, Patents 120
III. Synchronous Machines. Papers, Inventor’s Certificates, Patents 121
5 Representation of Currents in Rotor Short-Circuited Winding Elements in the Form of Generalized Characteristics 122
5.1 Problem Statement 122
5.2 Initial Data and Its Representation 124
5.2.1 Representation of Resulting Field Harmonics in Air Gap 124
5.2.2 Geometrical Dimensions of Winding Elements. Designation of Loop EMF 126
5.2.3 A.C. Resistances and Reactances, Currents in Winding Elements 126
5.3 System of Equations and Peculiarities of Matrix Structure of Its Coefficients 127
5.4 Solution Results: Currents in Elements of Short-Circuited Rotor Windings. Their Generalized Characteristics 129
5.5 Accounting of “Adjacent” Harmonics Fields by Means of Generalized Characteristics of Currents 132
5.6 Peculiarities of Numerical Realization of System of Equations for Calculation of Generalized Characteristics 133
Brief Conclusions 133
References 135
I. Monographs, general courses, textbooks 135
II. Induction machines. Papers, inventor’s certificates 136
III. Synchronous machines. Papers, inventor’s certificates, patents 136
6 Passive Quadripoles Recurrent Circuits of Various Structure: Investigation of Their Peculiarities for Modeling Process of Currents Distribution in Short-Circuited Rotor Windings
6.1 General Comments 137
6.2 Representation of Short-Circuited Rotor Winding Elements in the Form of Quadripoles and Recurrent Circuits 138
6.3 Passive Symmetrical and Asymmetrical Quadripoles 138
6.4 Structural Features of Passive Symmetrical and Asymmetrical Recurrent Circuits 139
6.5 Methods of Investigation of Passive Symmetrical Recurrent Circuits Described by “Step” or “Lattice” Functions 142
6.5.1 Difference Equations, Methods of Their Solution 142
6.5.2 Pecularities of Currents Distribution in Symmetrical Passive Recurrent Circuits 146
6.6 Open Passive Recurrent Circuits. Constants for Calculation of Currents Distribution Calculation Examples 147
6.6.1 General Comments 147
6.6.2 System of Equations for Constants 148
6.6.3 Calculation Examples 149
6.7 Investigation Methods of Passive Asymmetrical Recurrent Circuits 151
6.7.1 Difference Equations, Methods of Their Solution 151
6.7.2 Constants of Asymmetrical Passive Open Recurrent Circuit Their Determination
6.7.2.1 General Case of Asymmetry: {{/bf 0}} /lt {{/varvec /Delta}}{{/bf /rm Z}} /lt /infty 154
6.7.2.2 Asymmetry Limit Case: Circuit Break (?Z ? ?) 156
6.7.3 Calculation Examples 156
6.7.3.1 Asymmetry /Delta {{/bf Z}}_{{{{/bf N}}_{{{/bf P}}} }} = 2 157
6.7.3.2 Limiting Case 158
Appendix 6.1 158
Brief Conclusions 158
References 160
I. Monographs, general courses, textbooks 160
II. Asynchronous machines. Papers, inventor’s certificates, patents 160
III. Synchronous machines. Papers, inventor’s certificates, patents 160
7 Active Symmetrical and Asymmetrical Chain Circuits: Investigation of Their Peculiarities for Modeling Process of Currents Distribution in Short-Circuited Rotor Windings 161
7.1 Main Definitions. Structural Kinds of Active U-Shaped Chain Circuits and Peculiarities of EMF Distribution of Loops 162
7.2 Methods of Investigation of Active Symmetrical Chain Circuits with EMFs Changing Depending on Number of Link Under the Harmonic Law 164
7.2.1 Difference Equations, Methods of Their Solution 164
7.2.2 Constants for Currents in Active Open Symmetrical Chain Circuits 166
7.2.3 Constants for Currents in Active Closed Symmetrical Chain Circuits 167
7.2.4 Constants for Currents in Regular Closed Chain Circuits 168
7.3 Methods of Currents Investigation in Asymmetrical Active Open and Closed Chain Circuits 172
7.3.1 Method of Investigation of Currents in Active Open Chain Circuits with Asymmetrical (Damaged) Elements. Calculation Example 173
7.3.2 Method of Investigation of Currents in the Active Closed Chain Circuits with Asymmetrical (Damaged) Elements: Calculation Example 176
7.3.2.1 General Problem 176
7.3.2.2 Special Case—Circuit Break (/Delta {{/bf Z}} /to /infty ) 179
7.3.3 Modification of the Investigation Method of Currents in the Active Closed Chain Circuits with Asymmetrical (Damaged) Elements: Calculation Examples 180
Appendix 7.1 183
Appendix 7.2 183
Brief Conclusions [7] 184
References 185
I. Monographs, general courses, textbooks 185
III. Synchronous machines. Papers, inventor’s certificates, patents 185
8 EMF Induced by Resulting Field in Short-Circuited Loops of Damper Winding and Squirrel Cage 186
8.1 Initial Data and Its Representation 186
8.1.1 Representation of Resulting Field Harmonics in Air Gap 187
8.1.2 Initial Geometrical Dimensions of Damper Winding, Squirrel Cage, Pole Winding 188
8.2 Two EMF Components in Loops of Short-Circuited Rotor Winding 190
8.2.1 General Problem: EMF in Any Loop of Short-Circuited Winding 190
8.2.2 EMF of Damper Winding Loops Located on Pole 192
8.2.3 EMF of Damper Winding Loops Located on Cross Axis q 193
8.2.4 EMF of Loops in Squirrel Cage 194
8.2.5 Excitation Winding EMF 196
Brief Conclusions 197
References 199
I. Monographs, Textbooks 199
III. Synchronous Machines. Papers, Inventor’s Certificates, Patents 199
9 Investigation Methods of Currents Distribution in Regular Damper Windings and Squirrel Cages 200
9.1 Compliance Between Structures of Recurrent Circuits and Constructions of Short-Circuited Rotor Windings (Damper Winding, Squirrel Cages) 201
9.2 Symmetrical Squirrel Cage of Induction Machine [12, 13] 202
9.3 Two Definitions in Investigation of Currents in Squirrel Cage Loops Their Compliance and Areas of Correctness
9.4 Damper Winding of Salient Pole Machine [14–16] 208
9.5 Checking Results: Transformation of Expression for Currents in Elements of Complete Damper Winding of Synchronous Machine in Expression for Currents in Squirrel Cage Elements of Induction Machine 213
9.6 About the Determination of Damper Winding Reactances Based on Solution Obtained in This Chapter on Distribution of Currents in This Winding (Discussion Between L. A. Kilgore, Westinghouse El. Corp. and M. E. Talaat, Elliott Comp. [21, 22]) 215
Appendix 9.1: Method of Calculation of Overheats of Short-Circuited Rotor Winding Elements at Start-Up with Account for Change of the Main and Additional Losses in it from Temperature (with Account of Skin Effect) 218
Brief Conclusions 220
References 222
I. Monographs, Textbooks 222
II. Induction Machines. Papers, Inventor’s Certificates 222
III. Synchronous Machines. Papers, Inventor’s Certificates, Patents 223
10 Investigation Methods of Currents Distribution in Squirrel Cages with Asymmetry 224
10.1 General Comments 225
10.2 Currents in Asymmetrical Squirrel Cages of Induction Machine Rotors 225
10.2.1 Squirrel Cage with Asymmetrical Rotor Bar 226
10.2.2 Squirrel Cage with Two (Not Adjacent) Asymmetrical Bars in Rotor. Additional Currents in Winding Elements 226
10.2.3 General Problem: Squirrel Cage with Several (Not Adjacent) Asymmetrical Bars. Additional Currents in Winding Elements 227
10.2.4 Squirrel Cage with Three Adjacent Asymmetrical Bars (with Damages). Additional Currents in Winding Elements 230
10.2.5 Squirrel Cage with Three Asymmetrical Bars (with Damages): Two Bars Nearby, the Third—Next but One. Additional Currents in Winding Elements 230
10.2.6 Squirrel Cage with Three Asymmetrical Bars: Three Bars, not Adjacent. Additional Currents in Winding Elements 231
10.2.7 Calculation Example 232
Appendix 10.1: Additional Currents in Eq. (10.2) for Squirrel Cage with Two (Not Adjacent) Asymmetrical Bars 232
Appendix 10.2: Additional Currents in Eqs. (10.6) for Squirrel Cage with Three Adjacent Asymmetrical Bars 233
Appendix 10.3: Additional Currents in Eq. (10.7) for Squirrel Cage with Three Asymmetrical Bars: Two Bars Nearby, the Third—Next but One 234
Appendix 10.4: Additional Currents in Eq. (10.8) for Squirrel Cage with Three Asymmetrical Bars: Three Bars, Not Adjacent 234
Brief Conclusions 235
References 237
I. Monographs, Textbooks 237
II. Induction Machines. Papers, Inventor’s Certificates 237
III. Synchronous Machines. Papers, Inventor’s Certificates, Patents 238
11 Investigation Methods of Currents Distribution in Irregular Damper Windings 239
11.1 Currents in Damper Windings with Bars of Various Impedance on Each Pole 240
11.1.1 General Comments 240
11.1.2 Initial Data and Calculation Method for Currents in Winding Elements 240
11.2 Currents in Damper Winding with Damaged Bar on Pole 246
11.2.1 General Comments 246
11.2.2 Initial Data and Method of Calculating Currents in Winding Elements 247
Appendix 11.1 251
Appendix 11.2 251
Appendix 11.3 254
Appendix 11.4 254
Brief Conclusions 255
References 256
I. Monographs, textbooks 256
III. Synchronous machines. Papers, inventor’s certificates, patents 256
12 MMF of Damper Winding, Squirrel Cage (at Asymmetry in Them or at Its Absence) and Excitation Winding. Representation of MMF in the Form of Harmonic Series in Complex Plane 258
12.1 Fundamental Assumptions 258
12.2 Representation of Damper Winding and Squirrel Cage MMFs (at Asymmetry or Its Absence in Them) in the Form of Step Function 260
12.2.1 Initial Data and Their Representation 260
12.2.2 Peculiarities of Representing MMF and Field of Damper Winding and Squirrel Cage Currents in the Form of Step Function 261
12.3 General Method of Calculating MMF Harmonics and Fields of Rotor Currents Use of Symbolical Method of Representation of Currents in Combination with Complex Form of Harmonic Series Representation (Fourier)
12.3.1 Physical Treatment of Method. Calculation Expressions for Terms of Harmonic Series 265
12.3.2 General Expressions for Calculation of Complex Amplitudes of MMF Harmonics in Short-Circuited Winding of Arbitrary Construction 267
12.4 MMF and Field Harmonics in Squirrel Cage with Damages for Induction Machine (General Problem) 268
12.4.1 MMF Harmonics and Fields of Asymmetrical Squirrel Cage (with One Damaged Bar) Its Number Is N = NP
12.4.2 MMF Harmonics and Fields of Asymmetrical Squirrel Cage (Three Adjacent Damaged Bars) Their Numbers Are: N = 0, N = 1, N = 2
12.4.3 MMF Harmonics and Fields of Asymmetrical Squirrel Cage (Three Damaged Bars: Two Are Adjacent, the Third Is Next Nearest Their Numbers Are: N = 0, N = 2, N = 3)
12.4.4 MMF Harmonics and Fields of Asymmetrical Squirrel Cage (Three Damaged Non-adjacent Bars) They Numbers Are: N = 0, N = NP2, N = NP3
12.5 Harmonics of MMF and Fields of Symmetrical Squirrel Cage (Without Damages). Checking Results [9] 272
12.6 MMF Harmonics of Irregular Damper Windings [9] 275
12.6.1 First Construction Version of Irregular Damper Winding: Bars with Different Impedance Are Located Only on One or on Only Several Poles 275
12.6.2 Second Construction Version of Irregular Damper Winding: Bars with Various Impedances Are Located on Each Pole They Occupy Identical Position on Each Pole Relative to Its Longitudinal Axis
12.7 MMF Harmonics of Regular Damper Windings [9] 281
12.8 MMF Harmonics of Excitation Winding of Salient-Pole Machine and Screen on Polar Shoe [9] 282
12.8.1 MMF Harmonics of Excitation Winding 282
12.8.2 MMF Harmonics of Screen on Pole Shoe of Low-Speed Frequency Controlled Motor 283
Appendix 12.1: Accounting the Finite Width of Rotor Slots in the Calculation of Damper Winding MMF (Regular and Irregular) or Squirrel Cage (Symmetrical and Asymmetrical) 284
Brief Conclusions 285
References 288
I. Monographs, Textbooks 288
II. Synchronous Machines. Papers, Inventor’s Certificates, Patents 288
13 Field Harmonics in Air Gap of A.C. Machine in Nonlinear Network 289
13.1 Problem Setting 290
13.2 Initial Data and Their Representation 290
13.2.1 Complex Amplitudes (Phasors) of MMF Harmonics of Machine Rotor and Stator Loops and Their Representation 291
13.2.2 Equivalent Gap and Its Representation 291
13.2.3 Rotor Rotation Speed and Slip SSL 292
13.3 Calculation Method of Field Harmonics Excited by Rotor and Stator Winding MMFs 292
13.3.1 Induction Machines 293
13.3.2 Complex Amplitudes (Phasors) of Field Harmonics Excited by Damper Winding MMF and Excitation Winding MMF of Salient-Pole Synchronous Machine [13–15] 294
13.3.3 Rotation Speeds ?BOR,R in Air Gap of Field Harmonic Components Excited by Damper Winding MMF and Excitation Winding MMF of Salient-Pole Synchronous Machine 302
13.3.4 Complex Amplitudes (Phasors) of Field Harmonics Excited by Stator Winding MMF with Account of Cross Section Geometry of Salient-Pole Synchronous Machine at Rotor at Standstill (?REV = 0) and at Its Rotation (?REV?0) 304
13.3.5 Rotation Speeds ?BOR,ST in Air Gap of Field Harmonic Components Excited by Stator Winding MMF of Salient-Pole Synchronous Machine 306
13.3.6 Additional Ratios for Calculation of Complex Amplitudes (Phasors) of Stator Windings at |m| = |n| = |k| = 1, Their Check 308
Brief Conclusions [12] 309
References 311
I Monographs, textbooks 311
III. Synchronous machines. Papers, inventor’s certificates, patents 311
14 System of Equations for Magnetically Coupled Loops for A.C. Machine in Nonlinear Network 312
14.1 Problem Setting 312
14.2 Selection Method of Magnetically Coupled Loops for System of Equations 313
14.3 Features of System of Equations for Magnetically Coupled Loops 314
14.4 Basic System of Equations for Magnetically Coupled Loops 315
14.4.1 Formulation of System Initial Data and Results
14.4.2 Calculation Expressions for Flux Density Harmonics Complex Amplitudes Flux Density (Phasors) of Rotor and Stator Loops |m| = |n| = |k| = 1 317
14.4.3 Calculation Expressions for Complex Amplitudes of Rotor Winding Harmonics Flux Density at |m| = |n| = |k| = 1 317
14.4.4 Calculation Expressions for Complex Amplitudes of Stator Winding Harmonics Flux Density at |m| = |n| = |k| = 1 317
14.4.5 Result Summary: Calculation Expressions for Complex Amplitudes Field Speeds in Air Gap
14.4.6 Comparative Assessment of Separate Components of Complex Amplitudes (Phasors) 318
14.5 System of Magnetically Coupled Loops Equations [7, 8] 319
14.5.1 Equations for the First System of Loops Determined by EMF Frequency ?ST = Q1?1 319
14.5.2 Equations for the {{/varvec /upomega}}_{{{{/varvec ST}}}}^{{{{/varvec (2)}}}} {{/varvec = }}{{/varvec Q}}_{{{/varvec 2}}} {{/varvec /upomega}}_{{{/varvec 1}}} Second System of Loops Determined by EMF Frequency 320
14.5.3 Coupling Equations Between Both Systems of Stator Loops Determined by EMF of These Loops with Frequencies {{/varvec /omega}}_{{ST}} and {{/varvec /omega}}_{{ST}}^{ ( 2 )} 321
14.5.4 Peculiarities of Basic System of Equations [6] 322
Appendix 14.1: Accounting Higher Spatial Harmonics in a System of Equations of Magnetically Coupled Loops 323
Brief Conclusions 325
References 326
I. Monographs, Textbooks 326
II. Synchronous Machines. Papers, Inventor’s Certificates, Patents 327
15 Peculiarities of Operation Modes of A.C. Machine with Short-Circuited Rotor Windings at Nonsinusoidal Power Supply 328
15.1 General Comments 328
15.2 Peculiarities of Squirrel Cage Operation Mode (?REV  lessthan  ?1/p) 329
15.3 Peculiarities of Damper Winding Operation Mode (?REV = ?1/p) 330
15.4 Additional Measures to Decrease Damper Winding Losses in Salient-Pole Machine 331
Appendix 15.1 331
Brief Conclusions 332
References 333
I. Monographs, textbooks 333
II. Synchronous machines. Papers, inventor’s certificates, patents 333
III. State Standards (IE?, GOST and so on) 333
16 Operation Problems of High-Power AC Machines in Nonlinear Network 334
16.1 General Comments 334
16.2 Admissible Power of AC Machines in Nonlinear Network: Determination Methods Practical Examples [1, 11–13]
16.2.1 Losses in Stator Winding Carrying Alternating Current Containing a Number of Time Harmonics [1–4] 336
16.2.1.1 DC Losses 336
16.2.1.2 Additional Losses 337
16.2.1.3 AC Losses QAC,N 338
16.2.1.4 Losses in Stator Winding Carrying Sinusoidal Current of Industrial Frequency 338
16.2.1.5 Admissible Power of {/hbox{P}}_{{/rm w}}^{*} , Proceeding from Stator Winding Losses and Its Overheat 338
16.2.2 Losses in Machine Stator and Rotor Core Caused by Mutual Induction Field [2–6] Containing a Number of Time Harmonics 339
16.2.2.1 EMF and Current Frequencies in Machine Circuits 339
16.2.2.2 Mutual Induction Fields of Orders NAD and NDIR in Air Gap. Screening Factors SN,AD and SN,DIR 339
16.2.2.3 Losses in Stator Core 342
16.2.2.4 Losses in Rotor Core 343
16.2.2.5 Total Losses: Admissible Power Based on Losses in Machine Core and Its Overheats 344
16.2.3 Machine Admissible Total Power PADM 345
16.2.4 Calculation Examples: Determination of Machine Admissible Power PADM* 345
16.2.4.1 Calculation Peculiarities of DC and Additional Losses Based on (16.2?), (16.6), (16.9) for Rectangular Current Waveform (Limit Problem for Trapezium with Equal Top BTP and Bottom BBT Bases: BTP = BBT) 345
16.2.4.2 Calculation Results of Power {/hbox{P}}_{{/rm ADM}}^{*} for Trapezoidal Phase Current Waveform with Various Width of Top Basis (BTP = Var) Are Given in Table 16.1 346
16.2.5 Experimental Determination of Screening Factors SN,DIR, SN,AD of Asynchronous and Synchronous Salient Pole Machines 347
16.2.6 Checking of Rotor Short-Circuited Winding of Asynchronous and Synchronous Salient Pole Machines Heating Due to Higher Time Harmonics (N  greaterthan  1): Losses for Its Calculation Their Effect on Load of Operating Machines
16.3 Method of Determination Admissible Modes of High-Power Salient-Pole Generator Under Combined Load [11–13] 354
16.4 About the Level of Electromagnetic Load of Modern Salient-Pole Generators and Their Dynamic Characteristics in Independent Mode 357
16.4.1 General Comments 357
16.4.2 Requirements for Dynamic Modes 357
16.4.3 Transient Deviation of Generator Voltage ?U, Influence of Its Reactances 358
16.4.4 Inductive Resistances and Their Influence on Generator Weight and Dimensional Indicators 359
16.4.5 Problem Solutions: Offers 360
16.4.5.1 Offers for Newly Manufactured Generators 360
16.4.5.2 Offers for Generators Which Are in Operation 360
16.4.6 Additional Requirements to Generators 361
Brief Conclusions 361
References 363
I. Monographs, Textbooks 363
III. Synchronous Machines. Papers, Inventor’s Certificates, Patents 364
IV. State Standards (IE?, GOST and so on) 364
17 Frequency-Controlled Induction Motors in Nonlinear Networks: Assessment Criteria of Higher Harmonics Influence—Method of Criteria Calculation 365
17.1 Higher Harmonics and Need of Their Influence Assessment on Machine Modes in Nonlinear Network 365
17.2 Frequency-Controlled Induction Machines with Short-Circuited Rotor 366
17.2.1 Harmonics QAD  greaterthan  1 (m = 1): Slip SAD, EMF and Current Frequencies FAD,ROT in Rotor Loops. Power Balance in Secondary Loop 366
17.2.2 Harmonics QDIR  greaterthan  1 = (m = 1): Slip SDIR, Frequency of EMF and Currents in Rotor Loops. Power Balance in Secondary Loop 368
17.2.3 Technical and Economic Indicators Frequency-Controlled Induction Motors [22–26] 369
17.2.3.1 Quantitative Assessment Criteria of Higher Time Harmonics Influence 369
17.2.3.2 Assessment Methods: Offers and Their Substantiation 370
17.2.4 Calculation Peculiarities of Technical and Economic Indicators of Induction Motors in Nonlinear Network 372
Brief Conclusions 374
References 376
I. Monographs, textbooks 376
II. Induction machines. Papers, inventor’s certificates 376
III. Synchronous machines. Papers, inventor’s certificates, patents 377
IV. State Standards (IE?, GOST and so on) 377
18 Method of Minimizing Losses in High-Power Low-Speed Frequency-Controlled Motors in Operation Modes at Nonlinear Dependence of Shaft Torque on Rotation Speed 378
18.1 Application Areas of High-Power Low Speed Frequency-Controlled Motors 378
18.2 Voltage, MMF and Currents in Windings in Operation Modes (nREV  lessthan  nNOM) Mutual Induction Flux in Air Gap
18.3 Structure of Losses in Low Speed Frequency-Controlled Motor Efficiency in Operation Modes (nREV  lessthan  nNOM)
Appendix 18.1 382
Appendix 18.2 383
Appendix 18.3 383
Brief Conclusions 383
References 384
I. Monographs, textbooks 384
II. Synchronous machines. Papers, inventor’s certificates, patents 384
III. State Standards (IEC, GOST and so on) 384
19 Methods of Decreasing Nonlinear Distortion Factor in Voltage Curve of Salient-Pole Generator: Investigation of EMF Tooth Harmonics of Its Multiphase Winding with q per Pole and Phase as Integer 385
19.1 Introduction 385
19.2 Tooth Harmonics of Machine with q as Integer: Amplitude of Their Mutual Induction Field in Air Gap in no-Load Mode Frequency of This Field (Order of Tooth Harmonics)
19.2.1 Problem Formulation 387
19.2.2 Solution 388
19.2.3 Field of Harmonics bSLT(x,t,n) in Stator Slot Zone 389
19.2.4 Excitation Field bMI(x) (Rotor Field) Resulting Mutual Induction Field (in Air Gap)
19.2.5 Frequencies of Tooth Harmonics {{/varvec /omega}}_{{{/varvec Z}}}^{{({{/varvec 1}})}} /,{{/varvec and}}/,{{/varvec /omega}}^{{({{/varvec 2}})}} Calculation Expression for Nonlinear Distortion Factor
19.3 Rotor Construction with Local Shift of Poles in Tangential Direction 393
19.3.1 Peculiarities of Practical Realization 393
19.3.2 Tooth Harmonics of EMF eZ,0(t,n) in Stator Winding 394
19.3.3 Calculation Example 395
19.4 Rotor Construction with Group Shift of Poles in Tangential Direction 396
19.4.1 Peculiarities of Practical Realization 396
19.4.2 Tooth Harmonics of EMF eZ,0(t,n) in Stator Winding 397
19.4.3 Calculation Example 398
19.5 Generator Construction with Stator Axial Skewing of Stator Core or Rotor Poles 398
19.5.1 Peculiarities of Practical Realization 398
19.5.2 Calculation Example 399
Appendix 19.1 400
Appendix 19.2 400
Brief Conclusions 402
References 403
I. Monographs, Textbooks 403
II. Synchronous Machines. Papers, Inventor’s Certificates, Patents 404
III. State Standards (IEC, GOST and So On) 404
20 Methods of Decreasing Nonlinear Distortion Factor in Voltage Curve of Double-Fed Machines: Investigation of EMF Tooth Harmonics of Its Multiphase Stator and Rotor Windings with q Per Pole and Phase as Integer 405
20.1 Introduction 405
20.2 Assumptions 406
20.3 Peculiarities of Investigating EMF Tooth Harmonics of ASG Stator and Rotor Three-Phase Windings with Integer Number Q of Slots Per Pole and Phase 407
20.4 Rotor Fields of and Their Harmonics 407
20.4.1 Excitation Winding and Field of Its “Winding” Harmonics 407
20.4.2 Toothed Rotor Form Fields of Rotor Tooth Harmonics
20.4.3 Interaction of the First Harmonic of Mutual Induction Field and Field of Rotor Tooth Harmonics 409
20.5 The First Component of Resulting Mutual Induction Field in Air Gap of Tooth Order 409
20.5.1 Amplitudes of the First Field Component 409
20.5.2 Frequencies {{/varvec /upomega}}_{{{/bf Z}}}^{{({{/bf 1}})}} /hbox{,}/,{{/varvec /upomega}}_{{{/bf Z}}}^{{({{/bf 2}})}} {{/bf ,}}/,{{/varvec /upomega}}_{{{/bf Z}}}^{{({{/bf 3}})}} and {{/varvec /upomega}}_{{{/bf Z}}}^{{({{/bf 4}})}} of EMFs Induced in Stator Winding by the First Component of Field Harmonics 411
20.5.3 EMF Amplitudes {{/varvec e}}_{{{/varvec 1}}}^{{({{/varvec 1}})}} ,{{/varvec e}}_{1}^{{({{/varvec 2}})}} {{/varvec ,e}}_{{{/varvec 1}}}^{{({{/varvec 3}})}}/ {/hbox{/varvec and}}/
20.5.4 Nonlinear Distortion Factor {{/bf K}}_{{{{/bf DIST,1}}}} of ASG Stator Winding Voltage Caused by the First Component of Mutual Induction Resulting Field in Air Gap of Tooth Order 413
20.5.5 Calculation Example 413
20.6 The Second Component of Resulting Mutual Induction Field in Air Gap of Tooth Order 414
20.6.1 Amplitudes of the Second Field Component 414
20.6.2 Frequencies {{/varvec /omega}}_{{{/varvec Z}}}^{{({{/varvec 1}})}} ,/,{{/varvec /omega}}_{{{/varvec Z}}}^{{({{/varvec 2}})}} of the Second Component of Stator Winding EMF Tooth Harmonics 415
20.6.3 EMF Amplitudes {{/varvec e}}_{{{/varvec 2}}}^{{({{/varvec 1}})}} ,/,{{/varvec e}}_{2}^{{({{/varvec 2}})}} , Induced in Stator Winding by the Second Component of Pole Harmonics 415
20.6.4 Nonlinear Distortion Factor KDIST,2 of ASG Stator Winding Voltage Caused by the Second Component of Mutual Induction Resulting Field in Air Gap of Tooth Order 416
20.6.5 Calculation Example 416
20.7 The Third Component of Resulting Mutual Induction Field in Air Gap 417
20.7.1 Amplitudes of the Third Field Component 417
20.7.2 EMF Frequencies ?ST,W in Stator Winding Induced by “Winding” Harmonics of Rotor Field 417
20.7.3 EMF Amplitudes in Stator Winding Induced by “Winding” Harmonics of Rotor Field 418
20.7.4 Nonlinear Distortion Factor KDIST,3 of ASG Stator Winding Voltage Caused by the Third Component of Mutual Induction Resulting Field in Air Gap 418
20.7.5 Calculation Example 419
20.8 The Fourth Component of Resulting Mutual Induction Field in Air Gap of Tooth Order 419
20.8.1 Amplitudes of the Fourth Field Component Frequency EMF
20.8.2 EMF Amplitudes Induced in Stator Winding by the Fourth Component of Field Harmonics 420
20.8.3 Nonlinear Distortion Factor KDIST,4 of ASG Stator Winding Voltage Caused by the Fourth Component of Mutual Induction Resulting Field in Air Gap of Tooth Order 420
20.8.4 Calculation Example 421
Brief Conclusions 421
References 423
I. Monographs, Textbooks 423
III. Synchronous Machines. Papers, Inventor’s Certificates, Patents 423
IV. State Standards (IE?, GOST and so on) 424
21 Method of Determination Stator Winding MMF at Arbitrary Phase Current Waveform and Unequal Width of Phase Zones (for Investigation of Operational Characteristics of Frequency: Controlled Motors) 425
21.1 Introduction: Problem Formulation 425
21.2 Illustration of a Method of Solution for a Particular Case 428
21.2.1 Initial Data 428
21.2.2 Solution Stages 429
21.2.2.1 Current Time Harmonics 429
21.2.2.2 Symmetrical Current Components 429
21.2.2.3 Result: Stator Winding MMF Harmonics for Each System of Harmonics Currents at Q ? 1 and m ? 1 with Account of (21.7). Mutual Induction Fields 430
21.3 Solution of General Problem 430
21.3.1 Initial Data 431
21.3.2 Step Functions of Stator Current: Representation of Each of Them in the Form of Harmonic Series ?F(?, m, ?tS = idem) 432
21.3.3 Determination of Amplitude and Phase of Stator Current Step Function in the Form of Series ?F(?, m, ?tS = idem) of Spatial Harmonics of Order m 432
21.3.3.1 Peculiarity of Calculation Expression for MMF 433
21.3.4 Approximation of Stator Current Step Function Harmonics ?F(?, m, ?tS = idem) by Means of Time Harmonious Series Time Harmonics of Order Q ? 1
21.3.4.1 Conditions for Minimum Approximation Error Expressions for Calculation of Coefficients aQ, bQ of Harmonic Series
21.3.5 Result: MMF Harmonics of Stator Winding 437
21.4 Checking Results: Calculation Example 438
21.4.1 Phase Currents in Stator Winding 438
21.4.2 Three-Phase Six-Zone Single-Layer Stator Winding with Diametral Pitch and Phase Zones of Equal Width ?A = ?C? = ?B = ?A? = ?C = ?B? = = ?/3 Number q = 1. Position of Borders Between Phase Zones Along the Stator Periphery as Per (21.5): ?A = ?/3
Brief Conclusions 444
References 446
I. Monographs, Textbooks 446
II. Induction Machines. Papers, Inventor’s Certificates 446
22 Investigation Method of Transient Modes in Induction Machines with Rotor Cage Asymmetry 447
22.1 Introduction: Problem Formulation 447
22.2 Calculation of Currents Distribution in Asymmetrical Squirrel Cage Elements 448
22.2.1 Additional Aperiodic Current Components in Rotor Loops at Occurrence of Rotor Cage Asymmetry 448
22.2.2 Resultant Currents in Ring Portions /underline{/underline{{{/varvec I}}}}_{{({{/varvec N}})}} and Bars /underline{/underline{{{/varvec J}}}}_{{{{/varvec (N)}}}} After Breakage 450
22.3 Calculation of MMF for Asymmetrical Cage Currents 450
22.4 Calculation of Resulting Mutual Induction Fields (Fields in Air Gap) at n = p 451
22.4.1 Direct Field 451
22.4.2 Additional Field 453
22.5 Determination of Mutual Induction Factors and Parameters of Secondary Loops (Rotor) for the Main and Additional Fields 454
22.5.1 Mutual Induction Factor MDIR and Parameters of Secondary Loop (Rotor) for Direct Field 455
22.5.2 Mutual Induction Factor MADD and Parameters of Secondary Loop (Rotor) for Additional Field 455
22.5.3 Interrelation of Mutual Induction Factors and Parameters of Secondary Loops (Rotor) for Direct and Additional Fields 456
22.6 Equations for Calculation of Transients in Both Magnetically Coupled Machine Loops 457
22.6.1 Equations for Transient Currents in Stator ISTAT,DIR and Rotor ISEC,DIR Caused by Direct Field 457
22.6.2 Equations for Transient Currents in Stator ISTAT,DIR and Rotor ISEC,ADD Caused by Additional Field 458
22.7 The Resulting Transient Currents in Windings 460
22.7.1 Currents in Stator Winding 460
22.7.2 Currents in Secondary Loop (in Rotor) 460
22.8 Prospects for Using the Method with Account of Rotor MMF Higher Spatial Harmonics, and also for Calculations of Transient Currents in Synchronous Salient Pole Machines with Damper Winding 460
Brief Conclusions 461
List of symbols 462
References 464
I. Monographs, Textbooks 464
II. Induction Machines. Papers, Inventor’s Certificates 464
III. Synchronous Machines. Papers, Inventor’s Certificates, Patents 465
23 Theory and Methods of Investigation of Eddy Currents and Additional Losses in Stator Windings 466
23.1 Problem Formulation 467
23.2 Losses and Their Distribution in Bar Stator Windings 468
23.2.1 Winding Construction Features 468
23.2.2 Sources of Bar Losses: Fundamental Assumptions While Investigating Losses 470
23.2.3 Problem Setting and Solving Methods 471
23.2.4 General Problem of Eddy Currents and Losses Calculation in Elementary Conductors of the Bar 478
23.2.5 Bars with Solid Conductors (Complete Transposition): Calculation Example 484
23.2.6 Bars with Hollow Conductors (Complete Transposition): Calculation Example 485
23.2.7 Bars with Incomplete Transposition: Calculation Example 488
23.2.7.1 Currents Induced by Slot Leakage Fluxes 488
23.2.7.2 Derivation of Calculation Equations 490
23.2.7.3 Practical Choice of Elementary Conductors in Slot at Incomplete Transposition 497
23.2.8 Bars with Different D.C. Resistance of Elementary Conductors: Calculation Example 498
23.3 Numerical Methods of Eddy Current Investigation in Elementary Conductors of Bar Winding 508
23.3.1 General Observations 508
23.3.2 Problem Statement: Losses in Elements of Hollow and Solid Conductors of Slot Group 509
23.3.3 System of Equations for Currents in Elements of Slot Group Conductors. Circuits with Flux Linkage 511
23.4 Losses Distribution in Bar Winding 517
23.4.1 Additional Losses Distribution in Winding Turns in Slots 517
23.4.2 Losses in Several Bars of Double-Layer Winding 520
23.4.3 Losses Ratio in Outer Turns in Slot 521
23.4.4 Losses, Overheating and Bar Sizing in Non-standard Design of Stator Double-Layer Winding of Large Modern A.C. Machines 522
23.4.4.1 Introduction: Problem Statement 522
23.4.4.2 Calculation Equations 523
23.4.4.3 Practical Results Bar Sizing, Relations Between Bar Losses and Between Their Overheating 529
Bar Sizing and Relations Between Bar Overheating 531
23.4.4.4 Reduction of Losses in Both Winding Bars 534
23.5 Additional Losses in Coil Winding 537
23.5.1 Design Features 537
23.5.2 Fundamental Assumptions 540
23.5.3 Approaches to Solve the Problems of Additional Losses Caused by Slot Leakage Flux 540
23.5.4 Distribution of Circulating Currents 541
23.5.4.1 Special Case 541
23.5.4.2 General Case 544
23.5.5 Losses Due to Circulating Currents and Losses Distribution in Winding Turn 547
23.5.5.1 General Case 547
23.5.5.2 Special Case: SC ? 4 547
23.6 Questions for Self-Testing 549
Appendix 1: Method of Skin Effect Computation in Rotor Bars of Large Power Asynchronous Motor with Allowances for Temperature Distribution Therein 551
Appendix 2: Analytical Method of Designing Polyphase (mPH ? 3) Stator Winding with an Arbitrary Fractional Number Q of Slots Per Pole and Phase 558
Brief Conclusions [16] 561
References 562
I. Monographs 562
II, III. Induction and synchronous machines. Papers, inventor’s certificates 562
IV. State Standards (IEC, GOST and so on) 563